Imported Upstream version 8.2~beta3+dfsgupstream/8.2.beta3+dfsg

1 files changed, 86 insertions, 0 deletions

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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+(*                      Evgeny Makarov, INRIA, 2007                     *)
+(************************************************************************)
+
+(*i $Id: ZBase.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+
+Require Export Decidable.
+Require Export ZAxioms.
+Require Import NZMulOrder.
+
+Module ZBasePropFunct (Import ZAxiomsMod : ZAxiomsSig).
+
+(* Note: writing "Export" instead of "Import" on the previous line leads to
+some warnings about hiding repeated declarations and results in the loss of
+notations in Zadd and later *)
+
+Open Local Scope IntScope.
+
+Module Export NZMulOrderMod := NZMulOrderPropFunct NZOrdAxiomsMod.
+
+Theorem Zsucc_wd : forall n1 n2 : Z, n1 == n2 -> S n1 == S n2.
+Proof NZsucc_wd.
+
+Theorem Zpred_wd : forall n1 n2 : Z, n1 == n2 -> P n1 == P n2.
+Proof NZpred_wd.
+
+Theorem Zpred_succ : forall n : Z, P (S n) == n.
+Proof NZpred_succ.
+
+Theorem Zeq_refl : forall n : Z, n == n.
+Proof (proj1 NZeq_equiv).
+
+Theorem Zeq_symm : forall n m : Z, n == m -> m == n.
+Proof (proj2 (proj2 NZeq_equiv)).
+
+Theorem Zeq_trans : forall n m p : Z, n == m -> m == p -> n == p.
+Proof (proj1 (proj2 NZeq_equiv)).
+
+Theorem Zneq_symm : forall n m : Z, n ~= m -> m ~= n.
+Proof NZneq_symm.
+
+Theorem Zsucc_inj : forall n1 n2 : Z, S n1 == S n2 -> n1 == n2.
+Proof NZsucc_inj.
+
+Theorem Zsucc_inj_wd : forall n1 n2 : Z, S n1 == S n2 <-> n1 == n2.
+Proof NZsucc_inj_wd.
+
+Theorem Zsucc_inj_wd_neg : forall n m : Z, S n ~= S m <-> n ~= m.
+Proof NZsucc_inj_wd_neg.
+
+(* Decidability and stability of equality was proved only in NZOrder, but
+since it does not mention order, we'll put it here *)
+
+Theorem Zeq_dec : forall n m : Z, decidable (n == m).
+Proof NZeq_dec.
+
+Theorem Zeq_dne : forall n m : Z, ~ ~ n == m <-> n == m.
+Proof NZeq_dne.
+
+Theorem Zcentral_induction :
+forall A : Z -> Prop, predicate_wd Zeq A ->
+  forall z : Z, A z ->
+    (forall n : Z, A n <-> A (S n)) ->
+      forall n : Z, A n.
+Proof NZcentral_induction.
+
+(* Theorems that are true for integers but not for natural numbers *)
+
+Theorem Zpred_inj : forall n m : Z, P n == P m -> n == m.
+Proof.
+intros n m H. apply NZsucc_wd in H. now do 2 rewrite Zsucc_pred in H.
+Qed.
+
+Theorem Zpred_inj_wd : forall n1 n2 : Z, P n1 == P n2 <-> n1 == n2.
+Proof.
+intros n1 n2; split; [apply Zpred_inj | apply NZpred_wd].
+Qed.
+
+End ZBasePropFunct.
+